Hi PKTB :)
Read this excellent wifi thread from Ikehiker.
Wifi Guide for 2003SE
Offhand my experience with a situation such as yours is it may be the proxy settings.
Go into settings >connections >connections >set up my proxy server >check "This connects me to the internet box >tap General tab >your axim should have this already filled with
My Work Network.> tap the ok tab at the upper right of your screen.
The procedure I walked you through is to set your internet access for a Work connection. There is an excellent thread around here somewhere that explains the difference between Work and Internet. You'll be using the Work all the time! The Internet setting, if I remember, severely limits your connection.
Oh and if you're using WPA-PSK with TKIP data encryption. Good luck. My Axim has a very fickle relationship with our router here. I think you may have to set your router to a mixed signal and cross your fingers.
Good luck and welcome :)